In a small bowl, sprinkle yeast over 1/2 cup of the lukewarm milk. Stir to combine. Let sit until frothy, about 10 minutes.
In the bowl of a large food processor fitted with a dough blade or a large bowl, combine flour, salt, sugar, and lemon zest. Mix in the yeast with milk, eggs, and rum. Slowly add remaining milk, then butter until a soft dough forms. Transfer to a large bowl, cover, and let rest until doubled, about an hour.
Flour a large work surface and add dough. Roll the dough out until it is 1/2 inch thick. Cut out the doughnuts with a 2 1/2 inch circle cutter and place on baking sheet lined with parchment. Cover and let rise until doubled, 15-30 minutes.
While the doughnuts are rising, pour vegetable oil into a large pan until it is 1 inch deep. Heat the oil to 350 degrees F. Right before placing in oil, stick your thumb into the center of the doughnuts to create an indentation. Place a few, being careful not to overcrowd, indentation side down, into the hot oil and immediately cover with a lid.
Fry just until lightly browned, about 40 seconds. Remove lid and flip to fry other side just until golden, keeping lid off. Place on towel lined plate. Repeat with remaining doughnuts.
Serve sprinkled with sugar and topped with jam.